Problem 29 - College Physics

Question

Accident analysis. Two cars collide at an intersection. Car \(A\), with a mass of \(2000 \mathrm{~kg}\), is going from west to east, while car \(B\), of mass \(1500 \mathrm{~kg}\), is going from north to south at \(15 \mathrm{~m} / \mathrm{s}\). As a result of this collision, the two cars become enmeshed and move as one afterward. In your role as an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle of \(65^{\circ}\) south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car \(A\) going just before the collision?

Step-by-Step Solution

Answer

The enmeshed cars moved at approximately 7.33 m/s, and car A was going 11.55 m/s before the collision.

Step 1: Determine the Total Momentum After the Collision

The law of conservation of momentum states that the total momentum before the collision equals the total momentum after the collision. Since the cars move together after the collision, let their combined mass be \( 3500 \mathrm{~kg} \) (i.e., \( 2000 \mathrm{~kg} + 1500 \mathrm{~kg} \)). The velocity \( v_f \) of the enmeshed cars can be determined by breaking the momentum into components. We know the angle is \( 65^\circ \) south of east.Momentum in the x-direction: \( (3500 \mathrm{~kg}) \cdot v_f \cdot \cos(65^\circ) \).Momentum in the y-direction: \( (3500 \mathrm{~kg}) \cdot v_f \cdot \sin(65^\circ) \).Find the magnitude of the velocity using conservation of momentum.

Step 2: Apply Conservation of Momentum in the Y-direction

Since car \( B \) is the only car with initial momentum in the y-direction, \( m_B \cdot v_B = (1500 \mathrm{~kg}) \cdot (15 \mathrm{~m/s}) \).Thus the y-component of the momentum after collision is:\[ 3500 \cdot v_f \cdot \sin(65^\circ) = 1500 \cdot 15 \]Solve for \( v_f \):\[ v_f \cdot \sin(65^\circ) = \frac{1500 \cdot 15}{3500} \]Calculate \( v_f \).

Step 3: Calculate the Velocity Using the Y-Component

Calculate \( v_f \) from:\[ v_f \cdot \sin(65^\circ) = \frac{22500}{3500} \]This simplifies to:\[ v_f \cdot \sin(65^\circ) \approx 6.42857 \]Thus,\[ v_f \approx \frac{6.42857}{\sin(65^\circ)} \]Calculate \( v_f \).

Step 4: Determine the Initial Velocity of Car A Using X-Momentum Conservation

Initial momentum in the x-direction was only due to car \( A \):\[ m_A \cdot v_A = 3500 \cdot v_f \cdot \cos(65^\circ) \]Solving this, use the \( v_f \) determined above:\[ 2000 \cdot v_A = 3500 \cdot v_f \cdot \cos(65^\circ) \]Calculate \( v_A \).

Step 5: Calculate Car A's Initial Velocity

Solve the equation \[ 2000 \cdot v_A = 3500 \cdot v_f \cdot \cos(65^\circ) \] using the \( v_f \) determined previously. This provides you with the speed of car \( A \) just before the collision.

Key Concepts

Conservation of Momentum

In physics, momentum is a measure of the amount of motion an object has and is the product of its mass and velocity. The principle of conservation of momentum is crucial in collision analysis. It states that in a closed system, without external forces, the total momentum remains constant both before and after the collision.

In our exercise, two cars collide and become an enmeshed entity afterward. To apply conservation of momentum, we calculate the momentum for both vehicles before the collision and compare it to the momentum afterwards. Initially, car A moves in the east-west direction, while car B moves north-south. After colliding, they move together in a direction determined by their combined momentum.

• Before collision: Momentum is the sum of momentums of car A and car B separately.
• After collision: Total momentum is confined within their combined mass moving in a new trajectory.

Using the provided information and calculations ensures momentum components for both directions are conserved.

Velocity Calculation

Determining velocities involves breaking down the problem into steps and using mathematical components.

In this scenario, we want to find the resulting velocity after the collision and understand prior speeds. After the collision, the two cars become enmeshed, moving together. We begin by calculating the magnitude of their final velocity using momentum components and the provided angle of 65°.

Let's walk through the process:

• First, calculate the y-component of velocity since initial y-momentum arises only from car B.
• This involves rearranging the conservation equation, solving for the enmeshed cars’ velocity.
• Then, the x-component relates to car A, establishing another equation involving initial velocities.

Simplifying these equations results in us solving for the velocity of car A before collision and the final velocity after impact.

Momentum Components

Understanding momentum components is essential in dissecting collisions. Momentum acts in distinct directions, which is why we calculate both x and y components separately.

In this exercise, the enmeshed car momentum is explored along both axes:

• **X-direction**: Primarily involves the momentum of car A before collision. After collision, this component continues as part of the total momentum.
• **Y-direction**: Primarily involves car B’s initial momentum. It turns into the southward component of the total combined momentum.

By breaking momentum into these directions separately, scientists can reconstruct the movement accurately. Each component is addressed with trigonometric functions, utilizing angles provided to represent real-world directions.